I'm studying Poincaré Sections.
Every example I see, it calculates the solution $x(t)$ and then it uses it to calculate the Poincaré map. I don't see the point of the Poincaré map then. What information does the Poincaré map gives that having the explicit solution of the system doesn't? With that in mind, is it feasible to obtain a Poincaré map without calculating the solution of the system of equations?
